Preheat the oven to 350 degrees F (180 degrees C).
Spray four 1/2-cup ramekins with Vegetable Oil Cooking Spray (to taste) and arrange on a baking sheet, set aside.
Fill a medium saucepan 3/4 full with Water (to taste). Add Salt (1 teaspoon) and bring to a boil.
Slice the Chayote Squash (2) in half lengthwise and remove the seed. Add the chayote to the boiling water reduce heat to a simmer and cook for 30-35 minutes or until tender.
Transfer the squash to the bowl of a food processor and pulse several times to puree. If squash is very liquid-y and loose, pour it into a fine mesh sieve and strain the excess liquid from the puree. Transfer the puree to a bowl and set aside.
In a small saucepan over medium high heat, melt the Butter (1 tablespoon) and add the minced Shallot (2). Saute the shallots until tender and translucent.
Sprinkle in the All-Purpose Flour (1 tablespoon) and remaining salt and stir to coat the shallots, cooking for about 1 minute. Add the Milk (3/4 cup), stirring constantly and bring the mixture to a boil. Cook for one minute and remove from heat.
Stir in the chopped Fresh Thyme (1 teaspoon) and shredded Gruyère Cheese (1 cup). Add the béchamel sauce to the puree and stir to combine.
Divide the gratin evenly into the 4 ramekins and top with remaining cheese and Breadcrumbs (1/4 cup).
Bake for 20-30 minutes or until gratin is bubbly and breadcrumbs are toasted.
Let the gratin sit for 10 minutes before serving. Serve and enjoy!